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IN  MEMORIAM 
FLOR1AN  CAJORI 


JPROBLEMS  IN 

THE  TEACHING  OF 

SECONDARY  MATHEMATICS 


AN  ADDRESS  DELIVERED  BEFORE  THE 

NEW  ENGLAND  ASSOCIATION  OF 

TEACHERS  OF  MATHEMATICS 


BY 


DAVID  EUGENE  SMITH 


GINN  AND  COMPANY 

BOSTON  •  NEW  YORK  •  CHICAGO  •  LONDON 

COPYRIGHT.  1913,  BY  GINN  AND  COMPANY 


CAJORI 


$41 


PROBLEMS   IN  THE  TEACHING  OF 
SECONDARY  MATHEMATICS 

It  is  not  without  considerable  hesitancy  that  I  venture 
to  address  a  body  of  teachers  upon  some  of  the  great  prob- 
lems that  confront  us  at  the  present  time  in  the  work  in 
secondary  mathematics.  This  hesitancy  arises  from  several 
causes,  prominent  among  them  being  the  feeling  that  I 
shall  only  be  "  carrying  coals  to  Newcastle."  For  surely 
these  problems  are  already  in  your  minds,  and  many  of  you 
have  pondered  over  their  significance  and  their  solution 
quite  as  seriously  as  I,  and  no  doubt  with  a  more  satisfactory 
issue.  I  hesitate,  also,  because  I  can  merely  state  them  with 
no  attempt  at  solution,  mindful  all  the  time  of  the  ancient 
adage  referring  to  questions  which  a  wise  man  cannot 
answer.  But  after  all,  there  is  a  value  in  clearly  stating 
from  time  to  time  the  large  questions  that  confront  our 
guild,  for  if  problems  were  never  formulated  they  would 
never  be  solved,  and  it  is  upon  associations  like  this  that 
we  must  largely  depend  for  the  solution  of  those  that  I 
shall  venture  to  lay  before  you. 

I.    HAS  THE  DAY  OF  SECONDARY  MATHEMATICS 
PASSED? 

The  first  of  the  great  questions  that  confront  us  at  the 
present  time  relates  to  the  very  existence  of  secondary 
mathematics  in  our  curriculum.  To  many  of  us  it  may 


2  PROBLEMS  IN  TEACHING 

seem  preposterous  that  the  question  should  seriously  be 
asked.  We  say  to  ourselves  that  if  anything  is  to  be  blotted 
out,  let  it  be  some  language  or  one  of  the  various  manual 
arts  that  are  from  time  to  time  exploited  only  to  find,  in 
most  cases,  an  early  resting  place  in  the  great  educational 
necropolis  of  forgotten  graves. 

But  the  question  cannot  be  dismissed  in  any  such  way 
as  this.  Other  subjects  have  been  seemingly  as  well  in- 
trenched as  mathematics,  and  yet  they  have  passed  away. 
Formal  logic  was  at  one  time  one  of  the  great  features  of 
a  liberal  education  ;  it  gave  place,  in  the  secondary  school, 
to  formal  grammar ;  but  a  university  course  in  formal  logic 
is  now  a  rarity,  and  a  high-school  course  in  formal  gram- 
mar, as  it  was  conceived  of  a  few  years  ago,  is  almost 
unknown.  The  world  seems  to  proceed  as  well  without 
these  subjects  as  it  did  when  they  held  prominent  place, 
and  we  have  to  face  the  question  whether  it  would  not 
get  along  just  as  well  if  algebra  and  geometry  followed 
them  into  educational  oblivion.  The  medieval  Compotus 
was  once  an  essential  feature  in  the  education  of  a  learned 
man ;  it  was  apparently  intrenched  in  a  position  of  secu- 
rity ;  and  yet,  as  I  mention  it  to-day,  half  of  this  audience 
may  be  ignorant,  and  excusably  so,  of  even  the  meaning 
of  the  word.  Somebody  at  some  time  asked  the  question, 
"Why  should  an  educated  man  need  to  study  the  Com- 
potus ?  "  — and  the  answer  came  in  due  time,  "  There  is 
no  reason,"  and  the  subject  was  soon  forgotten.  Somebody 
to-day  raises  the  question,  "Why  should  an  educated  man 
need  to  study  algebra  ? "  —and  we,  the  teachers  of  mathe- 
matics, must  answer.  A  high  school  in  which  I  am  inter- 
ested holds  its  last  class  in  Greek  this  year  —  one  of  the 


SECONDARY  MATHEMATICS  3 

best-known  high  schools  in  the  country,  a  school  of  five 
hundred  selected  pupils,  all  of  whom  are  hoping  to  enter 
college.  Shall  it  be,  a  few  years  hence,  that  this  same 
school  shall  be  teaching  its  last  class  in  geometry,  com- 
pelled to  drop  another  ancient  and  honored  subject  because 
it  is  no  longer  demanded  ? 

We  may  say  to  ourselves  that  high-school  mathematics 
has  always  existed,  that  it  is  not  a  college  subject,  and  that 
it  is  absurd  to  talk  of  abolishing  it.  But  if  we  say  this  we 
forget  that  the  American  high  school  is  itself  a  new  insti- 
tution ;  that  it  has  no  exact  parallel  in  other  countries ; 
that  other  countries  select  their  pupils  for  secondary  work 
while  we  seek  to  educate  the  mass ;  that  95  per  cent  of 
our  high-school  pupils  do  not  go  to  college,  and  do  not 
hold  the  intellectual  standards  held  by  the  boys  of  our  old 
academies ;  and  that  it  is  comparatively  a  recent  demand 
of  the  American  college  that  its  candidates  must  offer 
any  mathematics  beyond  arithmetic.  So  when  we  speak  of 
high-school  mathematics  we  should  bear  in  mind  that  our 
high  school  has  not  as  yet  proved  its  worth,  and  is  even 
now  being  weighed  in  the  balance  with  rather  unsatisfac- 
tory results,  and  that  precollegiate  mathematics  is  only  a 
recent  matter.  If  a  pupil  postpones  his  Greek  until  he 
enters  college  to-day,  why  should  he  not  postpone  his 
Latin,  his  algebra,  and  his  geometry  to-morrow  ? 

Now  this  is  not  a  cry  of  alarm  for  the  sake  of  temporary 
effect ;  it  is  a  succinct  statement  of  the  arguments  that  we 
frequently  hear  to-day  from  the  general  educator.  Up  and 
down  this  country,  before  many  gatherings  of  teachers,  the 
question  is  being  vociferously  propounded,  "  Why  should 
the  girl  ever  study  algebra  ?  "  Even  in  associations  of 


4  PROBLEMS   IN  TEACHING 

teachers  of  mathematics  the  question  is  being  asked,  "  Who 
would  stand  to-day  for  the  spirit  of  Euclidean  geometry?  " 
And  the  men  who  ask  these  questions  hold  prominent 
positions ;  they  are  professors  in  universities,  educators  of 
influence,  men  whom  the  mass  of  teachers  naturally  look 
to  as  leaders.  The  problem  is,  therefore,  a  real  one  and 
one  that  we  have  to  face. 

But  we  must  not  deceive  ourselves  by  thinking  that 
we  can  successfully  meet  it  by  mere  opprobrium.  We  not 
infrequently  hear  it  asserted  that  the  general  educator  is 
usually  a  man  of  a  low  degree  of  scholarship,  that  his  range 
of  culture  is  limited,  that  he  was  taught  Latin  so  poorly 
that  he  believes  it  can  never  be  taught  in  any  other  way, 
and  that  he  rarely  stands  for  any  intellectual  ideals  ;  but 
we  must  remember  that,  even  if  the  assertion  has  some 
truth,  enough  people  of  this  type  might  easily  create  a 
Zeitgeist  that  would  not  down  by  any  such  formula  as 
Weave  a  circle  round  him  thrice."  These  men  who 
attack  the  ancient  culture  have  been  rather  recklessly  called 
educational  muckrakers,  men  who  seek  only  the  bad  and 
judge  everything  by  that ;  they  have  been  hysterically  de- 
nominated pedagogical  anarchists,  men  who  destroy  with- 
out rebuilding  ;  and  they  have  been  frequently  looked  upon 
as  intellectual  iconoclasts,  those  who  in  their  zeal  to  destroy 
idols  are  willing  that  all  the  beauties  of  art,  which  their 
dull  vision  fails  to  see,  should  go  to  the  scrap  heap.  In  all 
these  assertions  there  may  be  some  grain  of  truth,  but  no 
one  gains  anything  in  an  argument  by  giving  expression 
to  such  an  attitude  of  mind.  Such  an  expression  merely 
brings  the  countercharge  that  those  who  hold  other  views 
are  reactionaries,  laudatorcs  temporis  acti,  unprogressive, 


SECONDARY  MATHEMATICS  5 

and  selfish  clingers  to  their  little  jobs.    The  epithet  of 
"old  fogy"   is  as  weighty  as  that  of  "  muckraker." 

When  we  calmly  consider  the  question,  we  find  that  it 
relates  to  the  value  of  algebra  and  geometry  for  the  democ- 
racy that  America,  in  distinction  from  the  rest  of  the 
world,  is  trying  to  educate  in  the  high  school.  What  does 
democracy  want  of  mathematics  ?  And  in  our  America  of 
the  dollar  we  find  that  the  question  is  often  reduced  to  that 
of  the  immediate  utility  of  algebra  and  geometry.  The 
potential  utility  does  not  seem  to  enter  into  the  considera- 
tion of  the  type  of  reformer  that  seems  to  speak  the  loudest 
upon  the  subject.  And  here  appears  to  be  the  real  point 
at  issue  :  one  side  demands  the  immediately  useful,  while 
the  other  stands  for  that  which  it  claims  to  be  potentially 
so.  Can  we,  therefore,  justify  our  secondary  mathematics 
on  the  potential  side  ?  For  surely  no  one  would  for  a 
moment  claim  that  the  teaching  of  the  immediately  practi- 
cal part  of  algebra  to  a  mechanic  would  require  more  than 
a  month,  or  that  the  immediately  practical  part  of  demon- 
strative geometry  exists  at  all,  taking  these  words  in  their 
usual  popular  significance.  Such,  then,  is  the  first  of  the 
large  problems  that  seem  to  loom  up  before  us.  Call  it 
a  claim  for  mental  discipline  if  you  please  —  this  is  a  mere 
question  of  fashionable  or  unfashionable  phraseology ;  it 
is  a  claim  for  serious  attention  to  a  vital  issue  in  education. 

II.    WHY   SHOULD   NOT  ALL   MATHEMATICS   BE 
ELECTIVE? 

The  general  educator  is  usually  found  to  concede  that 
mathematics  should  be  taught  in  our  high  schools,  but  he 
is  frequently  heard  to  assert  that  it  should  be  elective. 


6  PROBLEMS  IN  TEACHING 

Many  teachers  of  mathematics,  perhaps  most  of  them, 
would  personally  welcome  such  a  change,  since  the  pleas- 
ure of  teaching  is  largely  increased  if  the  learner  takes  the 
subject  con  amore.  But  of  late  a  new  type  of  educator  has 
appeared,  the  one  who  proposes  to  weigh  in  psychological 
scale  the  intellect  of  youth  and  to  guide  it  aright.  You 
have  it  in  your  own  part  of  the  country  to-day  in  the  phase 
of  vocational  guidance,  and  in  this  work  so  many  excellent 
people  are  seriously  engaged  that  we  are  certain  to  see  it 
become  an  important  phase  of  modern  education.  Let  the 
boy  who  gives  promise  in  science  begin  his  specialization 
early,  say  those  who  seek  to  guide  the  youth  in  a  scientific 
manner,  and  let  the  one  who  takes  to  Latin  bend  his 
energies  there.  Let  there  be  scientific  tests  to  show  whether 
or  not  the  particular  individual  can  hope  for  success  in  the 
particular  vocation  —  a  worthy  effort  and  one  that  will 
produce  good  results.  But  there  are  not  wanting  those 
who  will  be  less  scientific,  and  who  will  assert  that  one 
who,  by  virtue  of  his  surroundings  and  family,  is  destined 
to  be  a  hewer  of  wood,  should  early  come  to  like  to  hew, 
and  should  be  taught  chiefly  the  nobility  of  labor  with  the 
hand.  That  we  may  realize  some  of  the  dangers  that  beset 
those  who  seek  to  guide  the  youth  aright,  and  who  may 
feel  called  upon  to  sidetrack  all  that  is  not  immediately 
practical,  let  me  tell  you  some  advice  that  I  myself  have 
given  within  a  few  years  past  in  cases  like  these,  and  lay 
before  you  the  problem  that  I  had  to  face. 

Not  long  ago  there  came  to  me  a  father  who  wished  to 
train  his  boy  for  trade  in  a  seaport  town,  and  who  asked 
my  advice  as  to  the  proper  education  to  give  him.  The 
problem  seemed  simple.  The  community  was  not  an 


SECONDARY  MATHEMATICS  7 

educated  one  ;  it  lived  off  its  little  shipping  industry  ;  the 
boy  was  destined  to  small  business  and  to  small  reward ; 
he  gave  no  promise  of  anything  better,  and  the  advice  was, 
therefore,  unhesitatingly  offered  that  the  only  mathematics 
he  needed  was  arithmetic  through  the  sixth  grade. 
"Another  parent  asked  me  a  little  later  about  his  son. 
The  boy  was  of  the  ordinary  type  and  would  probably 
follow  his  father's  occupation,  that  of  a  sculptor.  What 
mathematics  would  it  be  well  for  him  to  take  ?  I  suggested 
a  little  study  of  curves,  some  geometric  drawing,  and  the 
modeling  of  the  common  solids  —  a  bit  of  vocational 
guidance  that  seemed  to  me  then  and  seems  even  yet 
particularly  happy. 

A  third  boy  happened  to  be  with  me  on  a  steamer  and 
I  took  some  interest  in  talking  with  him  and  with  his 
mother.  They  lived  in  a  city  of  no  particular  note,  at  any 
rate  at  that  time,  and  the  boy  was  going  into  the  selling 
of  oil  within  a  few  years.  The  profits  of  the  Standard  Oil 
Company  appealed  to  the  family,  and  I  advised  him  to 
learn  his  arithmetic  well  and  get  into  business  as  soon 
as  he  could. 

Out  of  the  store  of  my  memory  I  recall  a  curious  lad 
whom  I  came  to  know  through  my  sympathy  with  the 
family.  The  mother  was  a  poor  woman,  and  she  took  the 
boy,  when  little  more  than  a  baby,  over  to  Riverside  Park 
one  day  when  there  was  a  naval  parade.  A  drunken  sailor, 
having  had  a  fight  with  a  group  of  hoodlums,  rushed 
through  the  crowd  of  spectators  and  slashed  right  and 
left  with  a  knife.  In  the  excitement  the  boy,  in  his  mother's 
arms,  was  horribly  cut  in  the  face.  When  I  knew  them 
he  was  about  ten  years  old,  unable  to  speak  plainly,  and 


PROBLEMS  IN  TEACHING 

already  a  misanthrope  through  his  affliction.  I  advised  the 
mother  to  give  the  boy  a  vocational  education,  telling  her 
that  through  the  use  of  the  hands  he  would  satisfy  his 
desire  for  motor  activity,  and  that  this  would  compensate 
him  for  the  loss  of  verbal  fluency  and  would  tend  to  make 
him  more  contented  with  his  lot.  In  this  advice  I  feel 
that  I  would  have  the  approval  of  educational  circles. 

And  finally,  out  of  this  series  of  experiences,  let  me 
recall  the  case  of  a  boy  whom  I  came  to  know  through 
a  noble  priest  who  found  him  one  morning,  an  infant  a 
few  days  old,  on  the  steps  of  his  church.  We  talked  over 
the  best  thing  to  do  for  such  a  foundling,  one  who,  at  the 
time  I  knew  him,  was  in  the  primary  grades.  He  showed 
no  great  promise,  he  was  without  family  recognition,  and 
his  only  chance,  apparently,  was  in  the  humbler  walks  of 
life.  I  recommended  a  vocational  school  where  he  could 
quickly  prepare  for  the  shop  or  the  lower  positions  of 
trade,  and  the  good  priest  approved  my  plan  at  the  time, 
although  he  finally  followed  quite  a  different  course. 

It  is  apparent,  however,  that  I  have  here  spoken  in 
parables.  Perhaps  you  already  recognize  the  boys,  and 
perhaps  you  feel  how  sadly  I  blundered  in  my  counsel. 
For  the  first  of  these  whose  cases  I  have  set  before  you 
felt  a  surging  of  the  soul  a  little  later,  and  this  was  recog- 
nized in  time,  and  he  became  one  of  the  Seven  Wise  Men 
of  Greece — Thales  the  philosopher,  he  who  introduced  the 
scientific  study  of  geometry  into  Greece.  The  second  felt 
a  similar  struggle  of  the  soul,  and  his  parents  recognized 
my  poor  counsel  in  time  to  save  him  and  to  give  to  the 
world  the  founder  of  its  first  university — Pythagoras  of 
Samos.  The  third  boy,  for  whom  only  .the  path  of  commerce 


SECONDARY  MATHEMATICS  9 

seemed  open,  and  this  in  a  town  only  just  beginning  to 
be  known,  was  the  man  who  finally  set  the  world's  first 
college-entrance  examination,  the  one  who  wrote  over  the 
portal  of  the  grove  of  Academos  the  words,  "  Let  no  one 
ignorant  of  geometry  enter  here  "  -  Plato,  the  greatest 
thinker  of  all  antiquity.  The  fourth,  the  hopeless  son  of 
poverty,  maimed,  sickly,  with  no  chance  beyond  that  of 
laboring  in  the  shop  for  such  wage  as  might  by  good  for- 
tune fall  to  his  lot,  became  the  greatest  mathematician 
of  his  day  —  always  the  stammerer  (Tartaglia),  but  one 
whom  Italy  has  delighted  to  honor  for  more  than  three 
centuries.  And  the  last  one  of  the  list,  the  poor  foundling 
on  the  steps  of  St.  Jean-le-Rond  in  Paris,  became  D'Alem- 
bert,  one  of  the  greatest  mathematicians  that  France,  a 
mother  of  mathematicians,  ever  produced. 

Shall  we,  then,  advocate  the  selection  of  those  who  are 
to  study  mathematics  and  close  the  door  to  all  the  rest  ? 
Are  we  so  wise  that  we  can  foresee  the  one  who  is  to  like 
the  subject,  or  succeed  in  it  ?  Have  we  so  adjusted  the 
scales  of  psychology  that  we  can  weigh  the  creases  in  the 
brain,  or  is  there  yet  invented  an  X-ray  that  will  reveal  to 
us  the  fashioning  of  the  cells  that  make  up  its  convolutions  ? 

Of  course  it  will  at  once  be  said  that  these  illustrations 
that  I  have  given  are  interesting,  but  that  they  are  unfairly 
selected ;  that  those  boys  gave  earlier  promise  in  mathe- 
matics than  I  have  said.  It  will  be  asserted  that  I  should 
have  taken  the  case  of  the  stupid  boy,  the  one  who  did 
not  like  school,  the  one  who  liked  to  play  with  little  wind 
wheels,  who  liked  to  fight,  who  actually  did  run  away 
from  school,  and  who  stood  near  the  bottom  of  the  class 
in  mathematics.  Such  a  case  would  be  a  fair  one,  one  in 


10  PROBLEMS   IN  TEACHING 

which  we  could  safely  say  that  prescribed  algebra  and 
geometry  are  out  of  place.  And  I  suppose  we  must 
agree  to  this  and  confess  that  the  argument  from  the  histor- 
ical incidents  that  I  have  mentioned  was  unsound.  Let  us 
rather  take  this  case  that  I  have  just  described,  and  let  us 
see  to  whom  the  description  applies.  I  need  hardly  tell 
you  who  this  boy  is  ;  he  is  well  known  to  you  ;  he  is  well 
known  to  the  world  ;  and  long  after  every  educational  re- 
former has  passed  into  oblivion  his  name  will  stand  forth  as 
one  of  England's  greatest  treasures,  for  it  is  the  name  of 
Sir  Isaac  Newton. 

But  again  I  have  been  unfair,  perhaps.  I  should  have 
taken  positively  hopeless  cases,  for  such  can  surely  be  found. 
I  should  have  taken  some  illiterate  man,  one  who  does  not 
learn  to  read  until  he  is  nearly  out  of  his  teens,  or  else 
some  man  who  shows  no  promise  in  mathematics  by  the 
time  he  reaches  manhood,  or  some  one  who  by  the  time  he 
is  thirty  is  to  show  no  aptitude  in  the  science.  It  is  so  easy 
to  theorize  !  But  let  us  have  care,  for  the  men  whom  I  have 
now  described  are  Eisenstein,  Boole,  and  Fermat.  Take 
them  away  and  wnere  is  your  theory  of  invariants,  your 
modern  logic  of  mathematics,  and  the  greatest  genius  in 
theory  of  numbers  that  the  world  has  ever  seen  ? 

But  I  am  wandering  afield,  and  I  fear  I  may  be  inter- 
preted to  question  the  modernizing  of  our  educational  work. 
I  thoroughly  favor  scientific,  vocational  guidance  as  under- 
taken by  a  small  number  of  our  best  scholars  in  the  field. 
With  equal  zeal  do  I  favor  industrial  education  when  it  is 
not  so  narrow  as  to  condemn  a  boy  to  some  particular 
groove  in  life,  and  I  earnestly  hope  that  we  shall  so  guide 
our  youth  that  every  boy  and  girl  will  leave  school  fitted  to 


SECONDARY  MATHEMATICS  1 1 

do  something  well.  I  do  not  believe  that  any  thoughtful 
educator  wishes  to  guide  a  pupil  in  a  narrow  path  nor  keep 
from  him  the  chance  that  the  world  owes  him.  It  seems 
right,  however,  to  set  the  problem  clearly  before  us  :  Can 
we  safely  say  that  we  may  close  the  door  of  mathematics 
to  any  boy  ?  Should  he  not  be  given  the  chance  ?  If  he 
fails,  that  ends  it,  but  if  he  succeeds,  the  world  is  the  winner 
in  the  lottery.  Of  course  this  does  not  answer  the  question 
as  to  what  this  chance  should  be ;  it  is  quite  possible  that 
it  should  not  be  our  present  algebra ;  it  is  even  possible 
that  it  should  be  merely  some  form  of  mensuration  that 
masks  under  the  time-honored  name  of  geometry  ;  it  may 
conceivably  be  some  emasculated  form  of  fused  mathematics 
that  has  none  of  the  logic  of  geometry  and  none  of  the 
beauty  of  algebra,  although  I  do  not  believe  it ;  and  it  may 
even  be  some  form  of  technical  shop  mathematics  that 
appeals  to  but  few  pupils  because  of  its  very  technicalities. 
This  is  the  part  of  the  problem  to  be  solved.  But  that  the 
door  of  mathematics  of  some  substantial  character  shall 
not  be  opened,  and  opened  after  arithmetic  has  been  laid 
aside  as  the  leading  topic,  seems  unthinkable. 

III.    DOES   THE   GIRL   NEED   TO   STUDY  ALGEBRA? 

A  third  question  that  seems  at  the  present  time  to  agitate 
.the  educational  interests  in  some  parts  of  the  country  relates 
to  the  study  of  algebra  by  the  girl.  This  carries  with  it 
the  corollary  that  no  mathematics  whatever,  beyond  mere 
computation,  is  to  be  required  of  at  least  half  the  force 
that  controls  the  world. 

How  this  meets  the  views  of  the  emancipated  woman  I 
do  not  know.  I  assume  that  she  would  say,  if  asked,  that 


12  PROBLEMS  IN  TEACHING 

if  algebra  and  geometry  are  good  for  the  boy,  save  in  the 
narrowest  technical  sense,  they  are  good  for  the  girl  also. 
I  should  think  she  would  say  that  if  mathematics  is  the 
one  subject  that  makes  us  understand  our  infinitesimal 
nature  in  the  infinite  about  us  ;  if  it  is  the  one  science  that 
has  had  the  most  to  do  with  banishing  the  superstition  that 
comes  from  simply  looking  at  the  heavens  with  lackluster 
eye  ;  if  it  brings  a  mental  uplift  that  no  other  science  brings 
and  lets  us  see  what  seems  the  nearest  to  exact  truth  of 
anything  that  we  meet ;  and  if  we  find  at  every  turn  the 
mathematical  invariant,  —  this  timely  symbol  of  the  un- 
changeable that  presents  itself  just  as  the  youth  is  passing 
into  manhood,  —  I  should  think  that  she  would  say  that 
if  mathematics  brought  these  things  in  its  train  it  is  worth 
while  for  the  girl  if  it  is  worth  while  for  the  boy. 

But  to  me  there  is  a  more  serious  side  to  the  problem. 
Mathematics,  in  one  form  or  another,  is  going  to  continue 
to  be  taught  in  the  schools.  It  will  gradually  change  to 
meet  the  future  demands  of  the  times  as  it  has  always  done 
in  the  past.  So  far  as  we  can  see  there  will  be  a  mathe- 
matics that  is  immediately  practical  for  those  who  are  not 
hoping  for  any  intellectual  leadership,  and  there  will  be  the 
mathematics  that  I  have  described  as  potentially  useful  for 
those  who  are  not  content  with  remaining  in  the  lower 
intellectual  class.  But  in  any  case  there  will  be  mathematics, 
and  the  boy  will  study  it.  The  question  as  it  relates  to  the 
girl  is,  as  I  have  intimated,  a  more  far-reaching  one.  When 
this  was  an  agricultural  country  the  father  directed  the 
education  of  the  children.  In  the  long  winter  evenings  he 
had  the  time  and  inclination  to  help  those  of  his  household 
who  at  that  season  of  the  year  were  taking  what  was  termed 


SECONDARY  MATHEMATICS  13 

their  schooling.  The  mother  had  other  duties  that  filled 
her  time,  and,  moreover,  was  not. herself  well  enough  edu- 
cated to  give  much  assistance  in  the  matter  of  study.  But 
America  has  changed.  With  our  urban  population  at  least, 
the  father  no  longer  has  control  of  the  education  of  the 
children.  In  our  manufacturing  centers  he  is  busy  in  the 
shop,  and  his  hours  are  no  longer  limited  by  the  light  of 
day.  On  the  other  hand,  the  urban  mother  no  longer 
weaves  and  spins,  no  longer  helps  in  the  fields,  no  longer 
preserves  fruit,  and  has  almost  forgotten  how  to  make 
bread.  She  has  more  time  for  the  higher  life,  and  it  is 
she,  rather  than  the  husband,  who  gives  the  direction^  the 
help,  and  the  encouragement  in  the  education  of  the  son 
as  well  as  the  daughter.  The  father  may  need  some  algebra 
in  his  trade,  for  if  he  reads  the  artisans'  journals  he  must 
know  how  to  manipulate  formulas,  but  "the  mother  must 
have  a  general  knowledge  of  the  subjects  that  children  study 
if  she  is  to  be  the  sympathetic  director  of  their  intellectual 
activities.  It  is  the  woman  quite  as  much  as  the  man  who 
needs  a  broad  education  at  the  present  time. 

When  I  hear  some  man  who  is  skilled  in  dialectics 
attack  the  teaching  of  algebra  to  the  girl  because  it  does 
not  enter  into  her  immediate  life,  I  am  led  to  wonder  if 
he  would  have  women  learn  anything  whatever  except  the 
work  of  the  household  and  the  club.  The  classics  of  the 
world  appeal  to  her  immediate  needs  and  interests  quite 
as  little  as  mathematics,  and  hence  these  would  be  dropped. 
Indeed,  we  hear  not  infrequently  of  some  school  principal 
who  proposes  the  epoch-making  plan  of  studying  literature 
only  from  current  magazines.  If  we  agree  to  drop  all  noble 
literature,  all  knowledge  of  history  except  what  comes  from 


14  PROBLEMS   IN  TEACHING 

the  newspapers,  all  study  of  good  music  and  art  except  by 
those  who  expect  to  be  musicians  or  painters  or  designers, 
all  science  except  such  as  relates  to  the  simple  chemistry 
of  foods,  all  astronomy,  all  physics,  everything  except  what 
concerns  the  science  of  good  physical  living,  then  we  may 
let  mathematics  go  with  the  rest.  Perhaps  this  is  not  what 
these  agitators  mean  —  one  never  knows,  for  they  them- 
selves never  seem  to  have  any  exact  ideas  upon  the  subject. 
Theirs  it  seems  to  be  to  agitate,  hut  never  to  formulate, 
and  yet  they  serve  a  purpose  in  the  economy  of  education  as 
a  yeast  microbe  does  in  the  economy  of  nature.  I  refer  here 
not  to  the  scientific  psychologist  who  is  seriously  working  on 
the  great  problem  of  education,  but  to  the  man  of  brilliant 
speech  who  prostitutes  his  powers  of  persuasion  to  cast 
doubt  on  whatever  fails  to  appeal  to  his  immediate  fancy. 
Hence  the  problem  seems  to  me  not  so  much  to  decide 
whether  or  not  the  girl  should  study  algebra,  as  to  decide 
how  we  shall  so  teach  the  subject  to  her  that  she  will 
know  of  its  beauties,  of  its  purposes,  and  of  the  feeling 
of  mastery  that  comes  from  its  pursuit.  Such  a  problem 
may  well  occupy  the  attention  of  associations  like  the  one 
I  have  the  honor  to  address. 

IV.    WHAT  SHALL  BE  THE  MATHEMATICS  OF 
OUR  TECHNICAL  SCHOOLS? 

A  fourth  problem  that  is  thrust  upon  us  by  modern  con- 
ditions relates  to  the  mathematics  of  our  technical  schools. 
While  we  do  not  hear  such  vociferous  assertions  as  we  did 
a  short  time  ago  about  the  fact  that  "the  doctrine  of  formal 
discipline  has  been  exploded,"  —a  resonant  but  rather 
uncertain  phrase  that  was  fashionable  for  a  time,  —  it  is 


SECONDARY  MATHEMATICS  15 

neverthless  quite  axiomatic  that  the  mathematics  of  a 
technical  school  should  aim  at  something  in  addition  to 
general  culture  or  power.  In  a  school  of  mechanics  we 
need  the  mathematics  of  mechanics,  and  so  for  other 
special  fields.  And  yet,  as  we  look  over  courses  of  study 
for  our  agricultural  schools,  for  example,  we  find  nothing 
but  arithmetic  taught  in  one  college,  while  mathematics 
through  the  calculus  is  taught  in  another ;  and  in  the  high 
schools  the  confusion  is  equally  apparent.  And  what  is 
true  for  agricultural  schools  is  quite  as  true  for  other 
technical  institutions.  We  have  not  even  begun  to  think 
seriously  about  solving  the  problem  for  our  high  schools, 
and  the  same  thing  can  be  said  for  our  industrial  schools  of 
a  more  elementary  character.  Mere  technical  mathematics 
alone  has  never  succeeded,  and  the  nature  of  the  general 
mathematics  that  is  best  suited  to  develop  the  power  to 
handle  the  problems  that  confront  the  foreman  who  is 
erecting  a  skyscraper  has  never  yet  been  determined. 
Here,  then,  is  another  serious  problem  that  meets  us  when 
we  try  to  settle  upon  the  best  course  in  mathematics  for 
the  technical  schools  of  our  country.  These  schools  are 
hardly  started  in  America  as  yet,  although  in  Europe  they 
are  well  established ;  and  if  we  may  judge  from  world  expe- 
rience, their  success  is  to  depend  in  no  small  degree  upon 
the  quality  of  mathematics  that  will  enter  into  their  curricula. 

V.    THE  PROBLEM  OF  OUR  BACKWARD 
AMERICAN  MATHEMATICS 

There  were  in  the  mind  of  those  who  initiated  the 
work  of  the  International  Commission  on  the  Teaching  of 
Mathematics  three  large  problems.  The  first  was  that  of 


16  PROBLEMS   IN  TEACHING 

leading  each  nation  to  take  stock  of  \t&  nwn  work  in 
presenting  mathematics  to  its  youth,  that  it  might  have 
before  it  a  kind  of  moving  picture  of  its  teaching,  from 
the  kindergarten  through  the  university.  The  second  was 
that  of  informing  other  nations  of  this  work,  to  the  end 
that  all  might  profit  by  the  success  and  failure  of  each. 
And  the  third  problem  was  that  of  looking  beyond  the 
confines  of  one's  own  land  and  seeing  what  the  rest  of  the 
world  is  doing.  In  the  United  States  upward  of  a  dozen 
reports  were  issued,  telling  the  story  of  our  own  work. 
But  there  remains  the  third  problem,  that  of  looking  abroad 
and  seeing  wherein  other  nations  are  surpassing  us,  and 
then  of  finding  the  causes  and  the  remedy.  This  is  the 
problem  of  the  immediate  future,  and  it  is  proper  on  this 
occasion  to  set  forth  one  of  its  phases. 

When  we  compare,  year  for  year,  the  work  in  mathe- 
matics here  and  abroad,  we  are  struck  by  the  fact  that  we  in 
the  United  States  are  not  only  not  the  leaders,  but  in  nearly 
every  respect  we  are  distinctly  behind  the  other  prominent 
countries  of  the  world.  At  the  end  of  our  seventh  grade 
we  are  about  a  year  behind,  and  at  the  end  of  our  twelfth 
school  year  we  are  about  two  years  behind,  other  great 
educational  nations  in  the  teaching  of  mathematics.  You 
say  this  to  a  professional  pedagogue  of  the  general  type 
and  he  will  make  all  sorts  of  apologies.  He  will  say,  "Oh, 
this  is  America,  and  we  have  different  problems," — as  if 
that  were  any  excuse.  He  may  lay  it  to  climatic  conditions, 
to  the  necessity  for  assimilating  a  million  immigrants  a 
year,  to  the  paucity  of  teachers  in  a  new  country,  to  the 
brief  tenure  of  office  of  women  teachers  and  even  of  men, 
to  our  shifting  population,  to  our  democracy  of  education, 


SECONDARY  MATHEMATICS  17 

to  the  greater  breadth  of  our  curriculum,  or  to  any  one 
of  dozens  of  other  causes.  But  he  is  a  rare  educator 
who  will  come  out  and  assert  that  we  have  a  soft  peda- 
gogy that  often  dominates  our  elementary  school,  a  sweet 
but  mushy  pedagogy  that  brings  a  maximum  of  temporary 
pleasure  with  a  minimum  of  intellectual  attainment.  Per- 
haps the  reason  that  he  does  not  say  this  is  that  it  is  not 
true.  Perhaps  we  are  wiser  than  our  European  neighbors 
in  selecting  topics  for  our  schools  that  are  better  adapted 
to  train  to  good  citizenship.  If  so,  the  results  certainly 
do  not  show  it  as  yet.  And  when  one  sees  the  vigor, 
contentment,  good  spirit,  and  comradeship  that  are  found 
in  this  generation  in  so  many  of  the  schools  that  represent 
the  influence  of  Pestalozzi  in  certain  of  the  countries 
abroad,  and  then  sees  the  intellectual  progress  that  these 
schools  foster,  he  must  question,  if  he  is  open-minded, 
the  wisdom  of  those  who  have  directed  the  American 
policy.  We  are  behind  ;  bad  as  the  European  arithme- 
tic is,  ours  is  worse ;  backward  as  the  European  boy  may 
be  in  his  algebra,  ours  is  more  so ;  faulty  as  may  be  his 
attainment  in  geometry,  that  of  our  boy  or  girl  is  still 
more  so.  I  know  of  plenty  of  schools  in  which  boys  at 
the  end  of  the  twelfth  school  year  have  a  good  working 
knowledge  of  trigonometry,  a  fair  command  of  the  basal 
principles  of  analytics,  and  enough  ability  in  the  calculus 
to  meet  the  demands  of  a  pretty  good  course  in  analytic 
mechanics,  but  they  are  not  American  schools.  I  join 
you  in  excusing  ourselves ;  I  know  the  standard  explana- 
tions by  heart ;  I  am  even  willing  to  join  with  every 
American,  of  the  type  satirized  by  our  foreign  friends, 
in  asserting  our  claim  to  having  the  longest  river,  the 


1 8  PROBLEMS  IN  TEACHING 

biggest  lake,  the  tallest  skyscraper,  the  wealthiest  men,  and 
the  most  abject  poverty  to  be  found  anywhere.  But  after 
all  this  boasting  is  over,  I  have  to  confess  to  myself  that, 
in  the  teaching  of  the  science  in  which  this  association 


and  I  are  interested,  America  is  behind,  definitely  and  un- 
questionably behind,  and  I  seek  not  for  a  dozen  causes 
so  much  as  for  one  good  remedy. 

When  we  attempt  to  free  ourselves  from  our  insular 
habit,  and  turn  our  attention  to  learning  from  the'  expe- 
rience of  the  rest  of  the  world  instead  of  from  the  theories 
of  the  lecture  room,  we  find  that  the  work  of  the  first  two 
years  is  generally  more  definite  in  other  countries  than  in 
ours.  The  idea  that  arithmetic  shall  be  merely  incidental 
in  these  years  is  not  held  abroad,  and,  indeed,  has  little 
scientific  standing  even  here,  although  it  has  numerous 
advocates.  Sixj^ears  are  elsewhere  generally  deemed  suf- 
ficient to  cover  the  essentials  of  arithmetic,  the  subject 
thereafter  being  reviewed  and  applied  along  with  algebra 
and  geometry.  Instead  of  beginning  formal  algebra  as  a 
new  topic  in  the  ninth  school  year  as  with  us,  the  subject 
is  introduced  by  easy  steps  in  the  sixth  and  seventh  grades, 
so  that  the  pupil  is  initiated  by  slow  degrees  into  the  ad- 
vanced stage.  Instead  of  withholding,  geometry  until  the 
tenth  school  year,  and  then  suddenly  springing  the  demon- 
strative "pKase  upon  the  pupil,  this  subject  is  introduced 
along  with  algebra  in  the  elementary  grades.  In  other 
words,  instead  of  devoting  the  seventh  and  eighth  grades 
to  a  business  arithmetic  that  is  often  too  difficult  for  the 
children,  a  simple  initiation  is  given  into  the  algebra  of 
the  formula  and  equation,  into  geometric  drawing  and  the 
simpler  demonstrations,  and  into  such  higher  arithmetic  as 


SECONDARY  MATHEMATICS  19 

is  within  the  grasp  of  the  pupils  of  these  grades.  There 
is  thus  an  intelligent  preparation  for  formal  algebra  and 
demonstrative  geometry  that  is  quite  lacking  with  us. 
Moreover,  in  the  eleventh  and  twelfth  school  years  there 
are  often  opportunities  to  take  courses  in  trigonometry, 
analytics,  the  calculus,  and  mechanics,  and  sometimes 
mathematical  astronomy,  that  are  almost  never  found  in 
our  schools,  and  that  have  proved  their  worth  by  the 
results  attained. 

It  would  be  a  sad  error  if  we  should  conclude  from  such 
a  statement  that  our  work  is  all  bad  and  the  foreign  work 
all  good.  The  human  tendencyjthat  Horace  satirized,  of  _. 
looking  on  our  neighbor's  possessions  as  better  than  our 
own,  must  always  be  recognized.  The  fact  is  that  we  have 
"IrT America  an  excellent  course  in  algebra  and  geometry, 
better  in  some  respects  than  those  found  abroad.  Our 
arithmetic  work,  too,  has  many  features  of  superiority.  But 
our  deficiency  seems  to  lie  in  three  features  :  our  dawdling 
over  early  primary  arithmetic,  our  neglect  of  the  initial 
stage  of  algebra  and  geometry  in  the  seventh  and  eighth 
grades,  and  our  failure  to  offer  advancedjel_ectiyes  in  the  . 
last  two  years  of  our  high  school. 

I  am  well  aware  of  the  difficulties  to  be  met  in  applying 
the  remedy,  and  it  is  unnecessary  to  dwell  upon  them  here. 
For  example,  I  know  the  gap  between  democracy  and  aris- 
tocracy in  education,  but  I  don't  believe  it  to  be  as  wide  as 
many  people  think.  I  wish  to  set  the  problem  before  you 
rather  than  to  attempt  to  point  out  the  slow  steps  by  which 
the  solution  may  be  effected.  If  an  ideal  is  kept  before  our 
people  we  will  all  gradually  move  toward  it,  and  this  gradual 
trend  is  wiser  and  safer  than  any  attempt  suddenly  to  attain 


20  PROBLEMS  IN  TEACHING 

results  that  seem  to  us  desirable.  The  problem  is  to  preserve 
the  serious,  orderly  mathematics  that  we  have,  while  adopting 
such  good  features  as  the  rest  of  the  world  may  suggest  to  us. 

VI.    THE  QUESTION  OF  PARALLELISM 

A  glance  at  the  foreign  schools,  such  as  has  been  given, 
suggests  another  problem  that  we  are  reasonably  certain  to 
meet  in  the  near  future  —  the  one  of  rearranging  our  cur- 
riculum. America  is  one  of  the  few  countries  in  which 
algebra  and  geometry  are  commonly  taught  in  tandem 
fashion.  The  required  work  in  most  of  our  high  schools 
is  one  year  of  algebra  followed  by  one  year  of  geometry, 
and  this  followed  by  an  elective  course  in  algebra,  and  this 
by  one  in  solid  geometry.  The  rest  of  the  world  in  general 
does  not  pursue  such  a  plan.  It  carries  its  algebra  and 
geometry  separately,  as  we  do,  but  "It  carries  them  side  by 
side,  say  from  the  seventh  grade  through  our  high-school 
^period.  _lt  may  definitely  assign  two  days  of  the  week  to 
algebra,  two  to  geometry,  and  one  to  arithmetic  ;  or  it  may 
make  some  other  arrangement,  but  in  any  case  the  arrange- 
ment is  based  upon  the  idea  of  parallelism.  I  do  not  believe 
that  America  is  ready  for  this  at  the  present  time.  Indeed, 
I  think  that  it  never  will  be  ready  unless  it  adopts  the  plan 
of  a  six-year  high  school,  beginning  with  the  seventh  grade, 
or  goes  even  farther  in  following  the  European  arrangement. 
But  there  are  many  arguments  in  favor  of  the  scheme  in 
case  such  an  administrative  change  is  made.  At  any  rate, 
the  plan  succeeds  everywhere  else,  and  because  it  does  not 
succeed  under  our  present  conditions  is  no  reason  for  believ- 
ing that  we  shall  not  have  to  face  the  problem  in  the  chang- 
ing conditions  that  are  likely  to  be  met  in  the  near  future. 


SECONDARY  MATHEMATICS  21 

VII.    ARE  WE  MAKING  MATHEMATICS 
INTERESTING? 

Among  all  of  the  experiments  that  have  been  made  in 
teaching  mathematics  in  this  country,  not  much  that  is 
strikingly  new  has  been  evolved,  nor  much  that  seems 
permanent.  We  have  had  in  general  education  a  great 
many  bubbles  to  prick,  as  witness  the  failure  of  the  type 
of  manual  training  that  was  a  few  years  ago  asserted  to  be 
a  panacea,  the  fading  away  of  the  doctrines  of  concentra- 
tion and  correlation,  the  semioblivion  into  which  the  cul- 
ture epoch  theory  has  passed,  and  the  fate  of  the  Grube 
method.  Likewise  in  the  teaching  of  mathematics  we  have 
our  bubbles,  bubbles  blown  with  the  enthusiasm  of  youth 
and  pricked  with  the  experience  of  years.  Sometimes  the 
bubble  is  a  geometry  syllabus,  sometimes  an  impossible 
fusion  of  mathematical  topics  as  far  apart  _as_  Latin  and~ 
chemistry,  sometimes  a  "ratio  method,"  sometimes  a  wild 
adoption  of  the  graph,  and  sometimes  a  form  of  practical 
mathematics  that  is  so  technical  that  it  repels  pupils  and 
teachers  alike.  And  hence  it  is  not  without  hesitancy  that 
I  suggest  what  very  likely  may  develop  into  another  bubble 
—  the  problem  of  making  mathematics  more  interesting  in 
and  for  itself. 

Although  no  two  sets  of  educational  statistics  ever  seem 
to  prove  exactly  the  same  thing,  all  statistics  that  touch 
upon  the  subject  tend  to  show  that  mathematics  ranks  well 
up  in  the  scale  of  pupils'  interests.  The  latest  set  that  I 
have  happened  to  see  placed  it  third  or  fourth  from  the 
top  on  a  scale  of  twelve,  with  the  vocational  training,  that 
was  to  have  been  our  salvation,  down  at  the  bottom  of  the 
whole  list.  However  taught,  mathematics  always  has  in  it 


22  PROBLEMS  IN  TEACHING 

the  game  element.  You  play  the  game,  and  you  win  if  you 
really  set  about  to  do  so,  and  when  you  win  you  have  a 
definite  result.  We  start,  then,  in  the  teaching  of  mathe- 
matics, with  this  great  advantage. 

But  the  question  arises,  Are  we  making  enough  of  this 
matter  of  interest  ?  If  we  give  a  couple  of  pages  of  dull, 
dry,  algebraic  formalism,  with  nothing  contributed  by  the 
teacher  to  enliven  it,  are  we  doing  our  best  ?  If  not,  what 
more  can  we  do  ? 

The  problem  is  not  one  to  be  solved  in  a  moment.  But 
let  me  suggest  that  the  history  of  mathematics  is  not  being 
used  as  wisely  as  it  might  be  in  our  classes.  I  do  not  think 
it  is  presented  to  best  advantage  in  the  form  of  discjorb_ 
nected  notes  in  a  textbook,  but  as  outside  material  to  be 
brought  into  class,  to  be  the  subject  of  a  moment's  inspir- 
ing talk  by  the  teacher,  it  has  unquestionable  value.  And 
so  it  is  with  the  recreations  of  mathematics,  a  subject  on 
which  we  have  considerable  available  literature.  Are  we 
using  this  material  as  we  should  ?  Is  it  feasible  in  our 
schools  to  establish  mathematical  clubs,  such  as  Mr.  New- 
hall  describes  in  his"  monograph  in  the  Commission  report 
on  mathematics  in  the  secondary  schools  ?  I  have  myself 
ventured  to  suggest  in  a  recent  number  of  the  Teachers 
College  Record,  under  the  title  "  Number  Games  and 
Number  Rhymes,"  a  few  possibilities,  and  to  incorporate 
some  material  on  the  subject  in  my  work  on  the  Teaching 
of  Arithmetic.  Certain  it  is  that  there  is  an  opportunity 
for  serious  work  here,  and  that  some  good  may  be  accom- 
plished if  we  do  not  go  to  the  extreme  of  making  a  mere 
bubble  out  of  the  effort.  There  would  not  be  so  much 
heard  about  the  immediately  practical  in  mathematics  if  we 

I 

,- 


SECONDARY  MATHEMATICS  23 

would  show  our  pupils  the  interest  in  the  subject  perse, 
and  the  meaning  of  the  science  in  the  larger  life  about  us. 
I  wonder  if  we  ourselves  ever  stop  to  think  of  the  effect 
of  blotting  out  of  existence  every  book  or  manuscript,  say 
even  on  so  important  a  subject  as  the  general  science  of 
education,  and  also  every  book  or  manuscript  on  mathe- 
matics. In  the  former  case  the  schools  would  open  next 
week  as  they  opened  this ;  the^teaching  would  go  on  as 
before ;  the  world  would  know  no  difference  save  in  a 
few  institutions  for  the  training  of  teachersI  and  even  they 
might  conceivably  be  the  betteroff .  But  in  the  case"~~tjf- 
mathematics  every  great  engineering  project  in  the  world 
would  be  stayed  ;  the  skyscraper  would  not  be  planned  ; 
the  next_ocean  leviathan  of  steel  could  not  be  begun  ;  the 
banking  of  the  country  would  halt ;  all  safe  navigation 
would  cease  ;  the  science  of  artillery  would  need  to  be 
begun  anew  ;  astronomy  would  stand  aghast ;  mechanics 
would  have  again  to  frame  its  laws  ;  and  civilization  would 
send  out  the  hurry  call  for  intellects  to  repair  the  damage. 
The  question,  then,  as  to  what  would  happen  if  mathe- 
matics were  taken  away,  might  well  be  suggested  to  a 
mathematics  club  in  a  high  school,  and  this  is  a  type  of 
dozens  of  other  questions  that  would  probably  add  to  the 
interest  that  the  pupils  take  in  the  study.  Imagine  the  joy 
of  a  healthy-minded  pupil  when  he  first  reads  "  Flatland.", 
or  "  Another  World,"  or  Hill's  "  Geometry  and  Faith,"  or 
when  he  comes  to  know  Ball's  "Mathematical  Recrea- 
tions,"  or  White's  "  Scrap  Hook  of  Mathematics  "  ! 

Sometimes,  however,  I  feel  that  we  get  little  thanks  for 
such  an  attempt.  I  know  a  certain  prominent  educator 
who  in  private  conversation  is  always  sane  and  thoughtful, 


24  PROBLEMS  IN  TEACHING 

but  who  now  and  then  seems  to  fall  into  grave  error  in 
his  public  utterance.  He  is  reported  as  having  recently  re- 
marked, in  one  of  his  somewhat  ill-considered  outbursts 
against  mathematics,  that  no  teacher  had  ever  yet  told  his 
daughter  why  she  studied  the  subject.  Of  course  it  is 
quite  probable  that  no  one  ever  told  her  why  she  studied 
anything.  He  admitted,  however,  that  she  was  very  fond 
of  the  subject,  but  he  raised  the  question  as  to  whether 
this  is  any  justification  for  teaching  it  to  her  or  to  any  one 
else.  I  know  of  no  one  who  would  make  any  such  claim, 
but  I  am  equally  certain  that  it  is  an  advantage  to  have 
the  subjects  that  we  teach  made  as  attractive  as  possible 
without  taking  from  them  the  sterner  qualities  that  they 
may  possess.  And  so,  at  the  risk  of  being  opposed  because 
I  take  such  a  commonplace  position,  I  suggest  as  one  of 
the  problems  of  our  time  the  effort  to  make  mathematics 
still  more  interesting,  independent  of  its  higher  purposes 
or  its  narrow  field  of  immediate  applications. 

VIII.    THE   FUNCTION   PROBLEM 

It  is  perhaps  well  to  inject  into  this  discussion  a  problem 
that  is  a  little  more  mathematical  than  the  ones  that  have 
been  suggested.  For  this  purpose  I  select  one  of  which  a 
great  deal  has  been  said  in  Europe  during  the  past  five  or 
ten  years,  and  one  of  which  we  are  soon  to  hear.  I  refer 
to  the  introduction  of  the  function  concept  early  in  our 
work  in  mathematics,  making  of  it  a  kind  of  unifying 
principle  of  the  elementary  science.  I  have  not  the  time 
to  refer  to  its  recent  history ;  to  the  rise  of  the  idea  of 
introducing  the  concept  into  elementary  work,  due  largely 
to  the  efforts  of  the  French  engineers ;  to  its  elaboration 


SECONDARY  MATHEMATICS  25 

by  men  like  Tannery ;  to  its  cool  initial  reception  by  the 
French  teachers ;  to  the  subsequent  efforts  of  Klein,  and 
to  its  recent  success  in  Germany.  Suffice  it  to  say  that 
there  seems  to  be  good  reason  for  bringing  before  the 
pupil,  as  soon  as  he  begins  the  study  of  algebraic  symbols, 
the  idea  of  function.  This  idea  naturally  enters  into  the 
study  of  the  simplest  formula,  it  is  the  essence  of  the 
equation,  it  is  the  major  part  of  mensuration  and  trigonom- 
etry, and  in  the  mathematics  that  follows  it  plays  a  part 
of  ever-increasing  importance.  The  question  is,  What  shall 
be  its  fate  in  America  ?  Shall  we  receive  it  with  Western 
enthusiasm,  exploit  it  as  we  did  graphs,  go  to  an  extreme 
from  which  we  must  recede  as  we  did  in  that  case,  and 
finally,  after  much  waste  of  energy,  come  down  to  a  sane 
use  of  this  valuable  aid  to  the  study  of  mathematics  ?  Or 
shall  we  first  ask  ourselves  the  question  as  to  why  we  study 
the  subject,  what  we  propose  to  accomplish  by  it,  where 
we  can  really  use  it  to  advantage,  and  what  are  the  extremes 
to  be  avoided  ?  If  we  take  the  latter  course,  we  shall  reach 
the  sane  stage  much  more  quickly  than  by  the  former 
route.  Already  the  danger  has  appeared,  and  so  it  is  well 
that  we  consider  this  as  one  of  the  problems  demanding 
our  serious  attention. 

IX.  THE  PROBLEM  OF  ALGEBRA  AND  GEOMETRY 

It  would  be  too  much  to  expect  you  to  listen  to  any 
extended  statement  of  the  problem  of  our  present  algebra 
and  geometry.  That  such  a  problem  exists  is  patent  to  us 
all.  That  conditions  are  not  ideal  we  would  all  agree.  It 
is  not  the  best  type  of  education  that  begins  the  work  in 
algebra  with  mere  formalism,  nor  the  work  in  geometry 


26  PROBLEMS  IN  TEACHING 

with  a  difficult  demonstration.  But  while  we  agree  to  this 
statement,  and  object  to  the  wretched  plan  that  was  at  one 
time  followed,  quite  as  strongly  as  our  colleagues  in  the 
general  field  of  education  can  possibly  object  to  it,  we 
must  recognize  the  fact  that  we  are  already  improving  the 
situation  very  rapidly,  quite  as  rapidly,  indeed,  as  seems 
safe.  Practically  all  of  our  better  algebras  to-day  have  in- 
troductory chapters  that  set  forth  the  uses  of  the  science 
in  a  simple  and  interesting  manner,  and  the  problems  that 
they  offer  are  becoming  more  real  from  year  to  year. 
Our  better  textbooks  in  geometry  no  longer  begin  with  a 
proposition  to  be  demonstrated,  but,  like  the  algebras,  have 
their  introductory  chapters  that  show  the  significance  of 
the  subject.  This  steady  progress  is  accompanied,  to  be 
sure,  by  many  amateurish  efforts  and  experiments  that  are 
foredoomed  to  failure,  but  the  great  body  of  common-sense 
teachers  is  not  disturbed  by  these  eccentricities  and  is 
steadily  progressing  to  a  better  understanding  of  the  prob- 
lem and  to  its  certain  solution. 

X.    THE  PROBLEM   OF  SCIENTIFIC  EDUCATION 

It  would  not  be  right  to  close  without  some  reference 
to  a  problem  that  has  recently  come  to  the  front  in  educa- 
tional circles,  fostered  to  no  small  extent  by  the  labors  of 
my  distinguished  colleague  Professor  Thorndike,  and  car- 
ried on  by  a  number  of  his  former  students.  I  refer  to 
the  attempt  to  find  scientific  sanction  for  what  we  do  or 
think  we  should  do  in  the'  classroom".  We  proceed  at 
present  as  a  result  of  what  we  believe  to  be  the  best  expe- 
rience of  the  race,  and  this  in  itself  is  a  scientific  method. 
But  may  we  not  accelerate  this  experience  and  improve 


SECONDARY  MATHEMATICS  27 

more  rapidly  ?  It  is  the  belief  of  most  educators  of  schol- 
arly rank  that  it  is  possible  to  find  out  the  causes  of  any 
unnecessary  retardation  in  the  advance  of  the  mass  of 
pupils,  and  we  should  certainly  encourage  all  scientific 
attempts  in  this  direction  so  far  as  they  relate  to  the  work 
in  our  field.  If  the  trained  investigator  can  show  us  where 
we  can  best  begin  informal  algebra,  informal  geometry, 
and  formal  mathematics ;  if  he  can  find  a  general  norm 
for  the  drill  work  that  shall  give  the  best  results ;  if  he 
can  tell  us  what  we  may  safely  demand  of  the  average  boy 
and  girl ;  and  if,  through  the  examination  of  a  sufficient 
number  of  cases,  he  can  standardize  the  work  in  mathe- 
matics in  our  high  schools,  we  shall  all  be  his  debtors. 
To  solve  this  problem  he  must  take  account  of  a  large 
number  of  factors,  and  his  efforts  will  not  meet  with  suc- 
cess until  these  factors  are  all  considered  ;  but  meanwhile 
we  should  recognize  the  problem  and  give  our  encourage- 
ment to  its  scientific  solution. 

XI.    THE  PROBLEM  OF  OUR  DUTY 

Addresses  like  the  present  one  are  ephemeral ;  they 
reach  no  conclusion,  and  they  are  not  likely  to  provoke 
very  serious  thought.  And  yet  they  serve  a  purpose  if  they 
select  from  the  wide  range  of  questions  of  the  day  a  few 
that  seem  to  demand  special  attention.  As  I  said  at  the 
outset,  I  have  no  definite  solutions  of  any  of  these  prob- 
lems, or  rather  I  do  not  believe  that,  by  laying  before  you 
such  as  I  have,  so  much  will  be  gained  as  by  stating  the  prob- 
lems themselves.  The  answer  to  all  such  questions  will  fol- 
low in  the  general  agreement  of  the  large,  silent,  thoughtful 
body  of  American  teachers.  But  it  would  not  be  right  that  I 


28  PROBLEMS  IN  TEACHING 

should  close  merely  with  the  enunciation  of  a  few  questions, 
accompanied  by  remarks  that  are  more  or  less  rambling. 
I  therefore  wish  to  suggest  one  more  problem,  that  of  our 
present  duty,  and  to  indicate  what  seems  to  me  its  solution. 
I  believe  it  to  be  our  duty  to  stand  solidly  against  the 
lowering  of  the  standard  of  mathematics  that  shall  make 
of  it  only  a  science  that  js  imjne^iaHy  ingtpaH  nf  poten- 
^tiallyjpractical.  I  believe  that  we  should  open  the  door  of 
the  great  field  of  algebra  and  geometry  to  every  boy  and 
every  girl  in  our  country.  If  they  fail,  let  them  substitute 
some  other  science  lor  this,  but  offer  them  the  chance.  If 
we  can  extend  our  high  school  downward  to  include  the 
seventh  grade,  the  offer  may  be  made  in  the  seventh  and 
eighth  grades,  and  this  may  be  better  than  to  require  the 
algebra  and  geometry  of  to-day.  But  to  close  the  door  of 
opportunity  in  any  one  of  the  great  branches  of  knowledge 
is  a  crime,  whether  it  be  in  biological  science,  in  music,  in 
language,  in  mathematics,  in  history,  or  in  any  other  field  of 
world  importance.  I  also  believe  it  to  be  our  duty  to  favor 
the  extension  of  high-school  mathematics  downward  to  the 
seventh  grade  at  least,  taught  by  the  high-school  teachers, 
which  teachers  should  in  the  near  future  be  trained  in 
physics  as  well  as  our  own  science.  With  this  should  come 
the  extension  of  electives  upward,  to  include  courses  in  an- 
alytics, calculus,  and  mechanics.  To  say  that  this  is  impos- 
sible is  to  say  that  the  American  youth  and  the  American 
teacher  lack  the  abilities  of  their  foreign  colleagues. 

I  believe  it  to  be  our  duty  to  encourage  in  every  way 
a  proper  industrial  education,  but  to  insist  that  serious, 
thoughtful  mathematics  shall  have  its  place,  and  shall  not 
become  merely  a  collection  of  a  few  rules  of  thumb.  I  feel 


SECONDARY  MATHEMATICS  29 

that  we  should  guard  against  making  the  function  concept 
ridiculous  as  we,  in  some  quarters,  made  the  graph  an 
absurdity ;  but  that  we  should  accept  it  for  what  it  is  really 
worth  and  use  it  wherever  in  our  teaching,  to  use  the  jargon 
of  the  pedagogue,  it  "functions."  I  conceive,  furthermore, 
that  an  association  like  this  is  unanimous  in  the  belief  that 
we  should  "  hold  fast  to  that  which  is  good,"  not  allowing 
ourselves  to  give  up  the  serious  study  of  algebra  because 
it  has  not  always  been  well  taught  in  the  past,  nor  the 
serious  study  of  demonstrative  geometry  because  some  one 
has,  for  the  millionth  time,  slaughtered  Euclid.  The  pos- 
sible improvements  that  are  suggested  by  the  problems  that 
I  have  ventured  to  lay  before  you  may  be  effected  with 
only  a  natural  and  gradual  change  of  the  science  as  we 
have  it ;  they  demand  no  cataclysm,  least  of  all  the  cata- 
clysm that  would  follow  if  some  of  our  reformers  had  their 
way.  When  we  see  a  man  claiming  that  he  speaks  for  the 
95  per  cent  of  high-school  pupils  in  demanding  the  aboli- 
tion of  real  mathematics,  we  will  do  well  to  listen  to  his 
objections,  trying  to  remove  any  just  cause  for  his  com- 
plaints. But  if  we  find  that  he  is  merely  the  advocate  of 
the  nonintellectual,  and  that  he  would  take  from  us  all  the 
idealism  that  we  have,  then  we  have  a  right  to  put  forward 
the  claims  of  the  boy  and  girl  who  really  wish  to  learn  and 
in  whose  souls  idealism  is  beginning  to  take  a  start.  But 
even  more  than  this,  we  have  the  duty  to  do  our  best  to 
bring  this  idealism  into  the  schools  for  the  95  per  cent  of 
whom  we  hear  so  much,  and  this  I  believe  we  shall  do 
effectively  only  when  our  high-school  work  begins  earlier 
than  at  present,  and  we  put  some  introductory  and  practical 
work  in  the  seventh  and  eighth  grades. 


30  PROBLEMS  IN  TEACHING 

And  above  all,  it  seems  to  me  to  be  our  duty  to  stand 
for  the  interest  of  mathematics  for  its  own  sake,  for  setting 
forth  its  beauty  of  symmetry,  for  voicing  its  poetry,  for 
living  its  religion,  and  for  exalting  it  because  of  the  truth 
that  it  sets  forth  so  clearly  and  because  of  the  invariant 
properties  that  characterize  it  in  every  branch.  So  far  as 
possible,  all  this  should  grow  out  of  the  child's  experiences 
and  needs  ;  not  merely  out  of  his  external  experiences  and 
needs  in  the  narrow  sense  of  the  workshop,  but  also  out  of 
those  which  are  more  vital  and  of  which  he  is  conscious 
within.  It  is  only  by  being  imbued  with  such  feelings  and 
ambitions  that  we  can  bring  our  pupils  to  love  the  subject 
and  to  feel  the  great  mental  uplift  that  comes  from  its 
study.  May  the  statement  of  the  few  problems  that  I  have 
set  before  you  assist  us  all  to  maintain  this  spirit  in  the 
future  as  I  am  sure  we  have  tried  to  maintain  it  in  the  past. 


